Introduction to Metric-Preserving Functions

Introduction to Metric-preserving Functions

Linear Functions Preserving Multivariate and Directional Majorization

Let V and W be two real vector spaces and let &sim be a relation on both V and W. A linear function T : V → W is said to be a linear preserver (respectively strong linear preserver) of &sim if Tx &sim Ty whenever x &sim y (respectively Tx &sim Ty if and only if x &sim y). In this paper we characterize all linear functions T : M_{n,m} → M_{n,k} which preserve or strongly preserve multivariate an...

Linear Functions Preserving Sut-Majorization on RN

Suppose $textbf{M}_{n}$ is the vector space of all $n$-by-$n$ real matrices, and let $mathbb{R}^{n}$ be the set of all $n$-by-$1$ real vectors. A matrix $Rin textbf{M}_{n}$ is said to be $textit{row substochastic}$ if it has nonnegative entries and each row sum is at most $1$. For $x$, $y in mathbb{R}^{n}$, it is said that $x$ is $textit{sut-majorized}$ by $y$ (denoted by $ xprec_{sut} y$) if t...

Introduction to Belief Functions

There are two major issues when it comes to dealing with belief formulation under ambiguity and uncertainty. In general, uncertainty deals with situations where one is not sure about the outcome of an event. For example, consider an urn with 100 balls, 50 red and 50 black. If one is considering the event of drawing a ball from the urn, the outcome of whether the color would be red or black is u...

Introduction to SAS Functions

A FUNCTION returns a value from a computation or system manipulation that requires zero or more arguments. And, like most programming languages, the SAS System provides an extensive library of “built-in” functions. SAS has more than 190 functions for a variety of programming tasks. This tutorial will cover the syntax for invoking functions, an overview of the functions available, examples of co...